Oscillation theory and numerical solution of fourth-order Sturm--Liouville problems |
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Authors: | GREENBERG LEON; MARLETTA MARCO |
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Institution: |
Department of Mathematics, University of Maryland, College Park Maryland, MD 20740, USA
Department of Mathematics, University of Leicester, University Road Leicester LE1 7RH, UK
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Abstract: | A shooting method is developed to approximate the eigenvaluesand eigenfunc-tions of a fourth-order Sturm-Liouville problem.The main tool is a miss-distance function M( ), which countsthe number of eigenvalues less than A. The method approximatesthe coefficients of the differential equation by piecewise-constantfunctions, which enables an exact solution to be found on eachmesh interval. In order to calculate N( ) for the approximateproblem, certain oscillation numbers NL and NR must be computed.These consist of sums of nullities (or rank deficiencies) of2 x 2 matrices obtained from the solutions of the approximatedifferential equation. Although these solutions can be foundexplicitly, the calculation of NL and NR is non-trivial, andis obtained by using certain properties of M( ). |
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